Solve for x fourth root of x^2-24x-3=0

$\sqrt[4]{x^{2} - 24 x} - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$\sqrt[4]{x^{2} - 24 x} = 3$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x^{2} - 24 x}}^{4} = 3^{4}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{2} - 24 x}$ as ${(x^{2} - 24 x)}^{\frac{1}{4}}$ .

${({(x^{2} - 24 x)}^{\frac{1}{4}})}^{4} = 3^{4}$

Simplify the left side.

Simplify ${({(x^{2} - 24 x)}^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${({(x^{2} - 24 x)}^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 24 x)}^{\frac{1}{4} \cdot 4} = 3^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(x^{2} - 24 x)}^{\frac{1}{4} \cdot 4} = 3^{4}$

Rewrite the expression.

${(x^{2} - 24 x)}^{1} = 3^{4}$

Simplify.

$x^{2} - 24 x = 3^{4}$

Simplify the right side.

Raise $3$ to the power of $4$ .

$x^{2} - 24 x = 81$

Step 4

Solve for $x$ .

Subtract $81$ from both sides of the equation.

$x^{2} - 24 x - 81 = 0$

Factor $x^{2} - 24 x - 81$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 81$ and whose sum is $- 24$ .

$- 27, 3$

Write the factored form using these integers.

$(x - 27) (x + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 27 = 0$

$x + 3 = 0$

Set $x - 27$ equal to $0$ and solve for $x$ .

Set $x - 27$ equal to $0$ .

$x - 27 = 0$

Add $27$ to both sides of the equation.

$x = 27$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 27) (x + 3) = 0$ true.

$x = 27, - 3$

Solve for x fourth root of x^2-24x-3=0

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